Question

consider the following curve.

$y = 3x^{2}-10x + 1$

find the slope $m$ of the tangent line at the point $(4,9)$.
$m=

find an equation of the tangent line to the curve at the point $(4,9)$.
$y=$

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Explanation:

Step1: Differentiate the function

The derivative of $y = 3x^{2}-10x + 1$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $y'=6x-10$.

Step2: Find the slope at the given point

Substitute $x = 4$ into $y'$. So $m=y'(4)=6\times4-10=24 - 10=14$.

Step3: Find the equation of the tangent line

Use the point - slope form of a line $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(4,9)$ and $m = 14$.
$y-9=14(x - 4)$
$y-9=14x-56$
$y=14x-47$

Answer:

$m = 14$
$y=14x - 47$